Scientific method.

How does science arrive at its results? Some people speak of the
"scientific method" as a set of "rules" for doing science. Too often such
rules are presented in schools as a "recipe" for doing science, and even
have numbered steps! That's misleading. At the other extreme, someone said
that scientific method is "Doing one's damndest with one's mind." I know
many have said better things about it, but here are some observations on
scientific method.

Science cannot answer all questions. Some questions have no answers.

How Science really works.

Even casual observation shows us that nature, as perceived by our senses,
has reliable regularities and patterns of behavior.
Through more precise and detailed study we found that many of these
regularities can be modeled, often with mathematical models of great
precision.

Sometimes these models break down when extended (extrapolated) beyond
their original scope of validity. Sometimes extrapolation of a model
beyond its original scope actually works. This warns us that we had better
rigorously test each model for validity, and these tests should be capable
of exposing any flaws in the modelflaws capable of demonstrating
that the model isn't true.

Even when a model survives such testing, we should only grant it
"provisional" acceptance, because cleverer people with more sophisticated
measuring techniques may in the future expose some other deficiencies of
the model.

When models are found to be incomplete or deficient, we often fix them by
tweaking their details till they work well enough to agree with
observations.

When rapid advances in experimental observations occur, a model may be
found so seriously inadequate to accommodate the new data that we may
scrap a large part of it and start over with a new model. Relativity and
quantum mechanics are historical examples. These situations are often
called "scientific revolutions."

When such upheavals occur, and old models are replaced with new ones, that
doesn't mean the old ones were totally "wrong", nor does it
mean their underlying concepts were invalid.
They still work within their scope
of applicability. Newton's physics wasn't suddenly wrong, nor were its
predictions found unreliable or incorrect when we adopted Einstein's
relativity. Relativity had greater scope than Newtonian physics, but it
also rested on a different conceptual basis.

Past experience has shown that mathematical models of nature have
tremendous advantages over the earlier, more appealing, models that use
analogies to familiar everyday phenomena of our direct sensory
experience. Mathematical models are less burdened with emotional baggage,
being "pure" and abstract. Mathematics provides seemingly infinite
adaptability and flexibility as a modeling structure.
If some natural phenomena can't be modeled by known
mathematics, we invent new forms of mathematics to deal with them.

The history of science has been a process of finding successful
descriptive models of nature. First we found the easy ones. As science
progressed, scientists were forced to tackle the more subtle and difficult
problems. So powerful are our models by now that we often delude ourselves
into thinking that we must be very clever to have been able to figure out how
nature "really" works. We may even imagine that we have achieved
"understanding". But on sober reflection we realize that we have simply
devised a more sophisticated and detailed description.

Whatever models or theories we use, they usually include some details or
concepts that do not relate directly to observed or measurable aspects of
nature. If the theory is successful we may think that these details are
matched in nature, and are "real" even though they are not experimentally
verifiable. Their reality is supposed to be demonstrated by the fact that
the theory "works" to predict things we can verify and continue to
verify. This is not necessarily so. Scientists often speak of energy,
momentum, wave functions and force fields as if they were on the same
status as objects of everyday experience such as rocks, trees and water.
In a practical sense (for getting answers) this may not matter. But on
another level, a change of scientific model may do away with a force field
as a conceptual entity, but it wouldn't do away with a forest, mountain
or lake.

Science progresses through trial and error, mostly error. Every new theory
or law must be skeptically and rigorously tested before acceptance. Most
fail, and are swept under the rug, even before publication. Others, like
the luminiferous ether, flourish for a while, and then their inadequacies
accumulate till they are intolerable, and are quietly abandoned when
something better comes along. Such mistakes will be found out. There's
always someone who will delight in exposing them. Science progresses by
making mistakes, correcting the mistakes, then moving on to make new mistakes. If we stopped making mistakes, scientific progress would stop.

What do scientists really think about 'reality'?

Scientists speak in a language that uses everyday colloquial words
with specialized (and often different) meanings. When a scientist
says something has been found to be 'true', what is meant isn't any form
of absolute truth. Likewise scientists' use of 'reality' and
'belief' don't imply finality or dogmatism. But if we inquire whether a
scientist believes in an underlying reality behind our sense impressions,
we are compounding two tricky words into a philosophical question for
which we have no way to arrive at a testable answer. I'd be inclined to
dismiss the entire question as meaningless, and not waste time discussing
it, or any other such questions. Yet a few scientists and
philosophers disagree, and wax eloquent in writing and speaking about
such questions.

The notion that we can find absolute and final truths is naive, but still
appealing to many people, especially non-scientists.
If there are any underlying "truths" of nature,
our models are at best only close approximations to themuseful descriptions
that "work" by correctly predicting nature's behavior. We are not in a
position to answer the philosophical question "Are there any absolute
truths?" We can't determine whether there is an underlying "reality" to be
discovered. And, though our laws and models (theories) become better and
better, we have no reason to expect they will ever be perfect. So we have
no justification for absolute faith or belief in any of them. They may be
replaced someday by something quite different in concept. At least they
will be modified. But that won't make the old models "untrue". All this
reservation and qualification about truth, reality, and belief, doesn't
matter. It isn't relevant to doing science. We can do science quite well
without 'answering' these questionsquestions that may not have any
answers. Science limits itself to more finite questions for which we can
arrive at practical answers.

Also, we've learned that not all questions we can ask have answers
that we can find. Any question that is in principle or in
practice untestable is not considered a valid scientific question. We
like to think that scientists don't waste time on those, but they seem to
pop up in discussion and in books quite often. (Many people think
unanswerable questions are the most profound and important ones. Questions
like "What is the meaning of it all," or "What jump-started the universe?"
I think that scientists should set these aside for the philosophers to
chew on, and get on with the business of answering more accessible
questions.)

Aesthetic appeal of theories.

Many who write about science emphasize the "beauty" and aesthetic appeal
of successful theories. I used to naively think that to achieve
intellectually and emotionally appealing theories was a goal of science.
Maybe it is, on the subconscious level, as a scientist may be more
enthusiastic about developing an appealing theory than an "ugly" one. And
if the appealing one "works" all the ugly alternatives are dropped and
forgotten.

But there's no reason why nature's operations should be beautiful or
appealing to us. There's no reason why nature's operations should even be
fully comprehensible to us. It could be that when we achieve an even more
successful theoretical description of nature it may turn out to be messy,
difficult to understand and use, and totally devoid of emotional or
aesthetic appeal. We may not be capable of devising more satisfying
alternatives.

We've had a taste of this already. When quantum mechanics was being
developed many physicists in the forefront of developing the theory
didn't "like" it, and hoped that someday they'd find a different way to
formulate the theoryone more to their liking. A couple of quotes
illustrate this:

Physics is very muddled again at the moment; it is much too hard
for me anyway, and I wish I were a movie comedian or something
like that and had never heard anything about physics!

Wolfgang Pauli (1900-1958) Austrian Physicist in the US.
(Nobel Prize, 1935). From a letter to R. Kronig, 25 May 1925.

In spite of great efforts to find a more appealing theory, and ingenious
attempts to show that such things as the Heisenberg uncertainty principle
were "wrong", the effort to remove the ugliness of quantum mechanics has
(so far) failed.

It seems almost inescapable that as physics becomes more successful and
more powerful its theories become farther removed from the intuitive,
simple, beautiful theories of earlier centuries. This shouldn't be
surprising. As we unravel the mysteries of the universe our first
successes are with those accessible to direct sensory
experiencephenomena that occur in everyday life and
are observable without specialized apparatus, phenomena
that have simple enough behavior that we can grasp the explanation and
feel we "understand" it. But now we have done all the simple stuff. So we
must sweat the details of phenomena that can't be directly sensed,
that can only be made to occur in the lab with expensive and sophisticated
equipment, and which require us to invent new mathematics to describe
what's happening. The fuel that motivates us to continue along these
lines is the fact that so often it works remarkably well, resulting in
both scientific and technological advances. The practical technological
fall-out from science stimulates funding of further research. But
inevitably the science upon which the technology of our daily lives
operates becomes farther removed from everyday experience and farther from
the understanding of non-scientists. Most people live in a world
that they understand only in a superficial way. That has been so since the
beginning of human history. Yet there was a time, in fairly recent
history, that almost anyone could feel that with a bit of effort and study
one could learn a lot more about science, and even have a feeling of
understanding much of science, and finding it intellectually and
emotionally satisfying. That is much harder to do today.

I think it was Von Neumann who said that if we ever make computers
that can think, with the power of the human brain or better, we won't know how
they do it. Future scientific advance may be carried out entirely by
computers, predicting phenomena of nature better than any previous models
and theories had. But the computers by that time will be evolving
independently of us, designing and re-designing themselves, learning
independently of our programmers, and finding their own algorithms for
dealing with nature. These algorithms will be so complex (and probably
ugly) that we won't know how they work, and won't be able to re-express
them in ways we can comprehend. One bit of evidence to show how this could
come about is the recent fuss over the Y2K (year-2000) problem. It's
terribly difficult to reconstruct the logic of computer programs written years ago, for which documentation is fragmentary, and the original programmers retired or deceased. Yet this problem is a small one compared to the problem of debugging a computer program written, not by a human, but by a computer that is redesigning itself as it works, in order to
solve problems that have frustrated the few greatest minds of humanity.

The symbiotic relation between mathematics and physics.

Students and laypersons seldom grasp the difference between mathematics and physics. Since math is the preferred modeling analogy for physics, any physics textbook is richly embellished with equations and mathematical reasoning. Yet to understand physics we must realize that math is not a science, and science is not merely mathematics.

In the early history of science, mathematics was considered a "science of
measurement", and was supported because of its practical applications in land measurement, commerce, navigation, etc. But those who did math discovered that mathematics was a branch of logic, and certain important results (such as the Pythagorean theorem of right triangles) could be arrived at by purely logical means without recourse to experiment. Slowly there emerged a body of knowledge called "pure" mathematicstheorems that were derived by strictly logical means from a small set of axioms. Euclid's geometry was of this form.

Scientists are explorers. Philosophers are tourists. Richard Feynman

Today science and mathematics are separate and independent disciplines. The physicist must learn a lot of mathematics, but the mathematician (unless working in an applied field) need not know science. In fact, most pure mathematicians seldom interact with scientists, and have no need to. Likewise, physicists generally are capable of doing mathematics without interaction with mathematicians, and have on a number of occasions, developed new mathematics to solve particularly knotty problems. One theoretical physicist I knew spent a lot of time reading the mathematics literature, saying "Those mathematicians are doing some stuff that might be really useful to us. I only wish they spoke our language."
His point was that the language with which each discipline speaks of its own field has diverged to the point where special effort must be made to "cross over" into the technical literature of the other field. A similar situation exists today in philosophy, where the language of philosophy of science has become so specialized and technical that most scientists find great difficulty reading it. But as one philosopher put it, "Philosophers of science observe scientists from outside, trying to figure out what they are doing, how they are doing it, and what it all means. In this process we have no need to talk to them. It's like watching a game where you don't know the rules when you come in, but try to figure out the rules by watching what the players do. For philosophers, science is a spectator sport."

For philosophers, science is a spectator sport.

Geometers can define concepts such as "circle", "triangle", "parallel lines". Within pure mathematics, these can be "perfect". The mathematician's
parallel lines are strictly and perfectly equidistant from each other, to a perfection unattainable by mundane measurement. All points of mathematician's circle are perfectly equidistant from its center, but no
one could draw such a perfect circle even with the best instruments.
The angles of a mathematician's triangle add to exactly
180°. But if you drew a triangle and measured the angles, each would have a finite precision and some experimental error, so the measured angles wouldn't add to exactly 180°, except accidentally.

By pure mathematics one can prove that the ratio of a circle's circumference to its diameter (called "pi") is approximately p = 3.1415927..., but we can also prove that
one cannot express it exactly with a finite number of decimal places. Its value is an unending decimalan irrational number. No measurement of real circles can have such perfect precision, so the value of p cannot be determined by experiment
on nature. This example illustrates that mathematics propositions cannot
be proven by experiment, only by pure logic. On the other hand, no scientific law or theory can be proven by using only the methods of mathematics.

The value of p is determined in the context of the axioms of Euclidean geometry. Mathematicians have also devised other, non-Euclidean, geometries. How do we even know that our universe conforms to Euclidean geometry? Measuring mechanically drawn circles is not useful for this, for they are too small. But we can test the geometry of space in subtler ways, and we have determined that Euclidean geometry is at least approximately true in our own cosmic neighborhood, and also out to
very great distances that astronomers have observed. For "local" measurements, space is closer to Euclidean than the precision of our best measuring instruments. If we were to do such measurements and found that the angles of a triangle consistently added to something larger or smaller than 180° we would conclude that space was curved. If that were so, the value of the ratio of a circles circumference to its diameter would be larger or smaller than the value p computed from Euclidean mathematics. We would also have to inquire about the practical physical meaning of "straight" as in the "straight" sides of a triangle, or the path of a ray of light in a vacuum.

Mathematics is a handy analogy that can be used to model parts of nature.
The mathematics can be carried out to whatever precision is needed, or "good enough" for a particular scientific purpose. Mathematics cannot discover new scientific truths, but as we develop science through hypothesis testing, mathematics can not only test the hypotheses against measurements, but help us refine (tinker) the hypothesis to bring them in closer agreement with experiment.

Logical deduction, including mathematical logic, is the language with which we frame our theories of physics. Mathematics is capable of
far greater power and precision than mere words. In fact, it is the language in which may physicists do their creative thinking.
It is also the tool we use to test our theories against the
final (and unforgiving) arbiter of experiment
and measurement. But mathematics is not a royal road to scientific truth.